Markov partition and Thermodynamic Formalism for Hyperbolic Systems with Singularities

Abstract

For 2-d hyperbolic systems with singularities, statistical properties are rather difficult to establish because of the fragmentation of the phase space by singular curves. In this paper, we construct a Markov partition of the phase space with countable states for a general class of hyperbolic systems with singularities. Stochastic properties with respect to the SRB measure immediately follow from our construction of the Markov partition, including the decay rates of correlations and the central limit theorem. We further establish the thermodynamic formalism for the family of geometric potentials, by using the inducing scheme of hyperbolic type. All the results apply to Sinai dispersing billiards, and their small perturbations due to external forces and nonelastic reflections with kicks and slips.

A piecewise smooth Fermi–Ulam pingpong with potential

“…Proof of Theorem 8. Under the assumptions 1-5, the limit map F∞ satisfies the conditions of [3] and thus it admits a Young tower with exponential tail. We recall from §3.2 that F∞ well approximates the original collision map F at infinity.…”

Polynomial Decay of Correlations for Flows, Including Lorentz Gas Examples

“…The first return map in the examples considered here is nonuniformly hyperbolic, modelled by a Young tower with exponential tails, so [27] does not apply directly. In a recent preprint, [10] have announced the existence of a uniformly hyperbolic first return. This combined with [27] may yield the asymptotic (9.19).…”

A Piecewise Smooth Fermi-Ulam Pingpong with Potential

“…Proof of Theorem 8. Under Assumptions 1-5, the limit map F∞ satisfies the conditions of [3], thus it admits a Young tower with exponential tail. We recall from Section 3.2 that F∞ well approximates the original collision map F at infinity.…”